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9    MEASURABLE FUNCTIONS       143

directly (13.5.5) that, for any mapping g : X -* R, we have fi*(g) = v*(g \ Y),
and hence that such a mapping g is /j-integrable if and only if g \ Y is v-
integrable, in which case we have

ftolYX.-f,

J                    J

(13.9.20.2)

Likewise, a mapping u of X into a topological space Z is //-measurable if
and only if u \ Y is v-measurable.

(13.9.21)   If n : X -+ X' is a homeomorphism, then a mapping u' of X' into a
topological space Y is 7i(/i)-measurable if and only if u o n is ^-measurable.

PROBLEMS

1.    Let (fmn) be a double sequence of /x-measurable mappings of X into a metric space Y.
Suppose that for each m the sequence (fmn)n-*i converges almost everywhere to a
function gm, and that the sequence (gm) converges almost everywhere to a function h.
Show that there exist two strictly increasing sequences of integers (mk), (nk) such that
the sequence (fmknk)kzi converges almost everywhere to h. (First prove the result when
X is compact by observing that, for each e> 0, there exists a compact subset K c X
such that jit(X  K) <> e and such that the sequence (gm) and each of the sequences
(/mn)n=si are uniformly convergent on K. Then pass to the general case by using the
diagonal trick.)

2.    (a)   Let/, g be two real-valued functions ^0, and suppose that g is /z-measurable.
Show that ju,*0^) = inf ^(ug), where u runs through the set of all /^-measurable
functions ^/.

(b) Let/, g, h be three functions ^0 on X, finite or not, and suppose that g and h
are /^-measurable. Show that p*(f(g 4- h)) = fi*(fy) -f ft*(//z) (using the convention
of (13.11) for the product of 0 and + o).

3.    Let/: X -> R be /x-measurable. For each n E Z, let An be the set of all x e X such that
2"-1 < |/(*)| g 2". Show that/is integrable if and only if 2 2>(An) < + oo.

n 6 Z

4.    Let (/) be a sequence of integrable functions on X, converging simply to a function/,
(a)   If/is integrable and \ fdp.= lim    / dp, show that for each e > 0 there exists

J                          JT-+-00   J

an integrable set A, an integrable function g J> 0 and an integer 0 such that, for all
J>/z0, we have          fnd^ <Je and \fn(x)\ g \g(x)\ for all xeA. (Consider an

integrable set B such that        |/| dp <! and such that/is bounded on B, and apply

J X B

EgorofT's theorem.)ible set N, such that u \ Kn is continuous for all n ; hence (c)